The Art of Interface

# arcsec — trigonometric arc secant function

Category. Mathematics.

Abstract. Trigonometric arc secant: definition, graph, properties, identities and table of values for some arguments.

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

## 1. Definition

Arc secant is inverse of the secant function.

## 2. Graph

Arc secant is discontinuous function defined on entire real axis except the (−1, 1) range — so, its domain is (−∞, −1]∪[1, +∞). Function graph is depicted below — fig. 1. Fig. 1. Graph of the arc secant function y = arcsecx.

Function codomain is limited to the range [0, π/2)∪(π/2, π].

## 3. Identities

Complementary angle:

arcsecx + arccscx = π/2

and as consequence:

arcsec csc φ = π/2 − φ

Negative argument:

arcsec(−x) = π − arcsecx

Reciprocal argument:

arsec(1/x) = arccosx

Sum and difference:

arcsecx + arcsecy = arcsec(xy /{1 − xy√[(1 − 1 /x2)(1 − 1 /y2)]})
arcsecx − arcsecy = arcsec(xy /{1 + xy√[(1 − 1 /x2)(1 − 1 /y2)]})

Some argument values:

Argument xValue arcsecx
10
√6 − √2π/12
√(50 − 10√5) /5π/10
√(2 − √2)π/8
2√3 /3π/6
√5 − 1π/5
√2π/4
√(50 + 10√5) /53π/10
2π/3
√(4 + 2√2)3π/8
√5 + 12π/5
√6 + √25π/12
Table 1. Arc secant for some argument values.

## 4. Support

Trigonometric arc secant function arcsec is supported in:

Trigonometric arc secant function of the complex argument arcsec is supported in:

## 5. How to use

To calculate arc secant of the number:

``arcsec(-1);``

To calculate arc secant of the current result:

``arcsec(Rslt);``

To calculate arc secant of the number x in memory:

``arcsec(Mem[x]);``